Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles.
• It is fundamental to fields like geometry, navigation, physics, and engineering.
• Trigonometry primarily focuses on right-angled triangles, but its principles can be extended to all triangles.

Basic Trigonometric Ratios

• The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
• These ratios relate the angles of a right-angled triangle to the lengths of its sides.
• Consider a right-angled triangle with one angle labeled θ (theta).
• The side opposite to θ is called the "opposite" side.
• The side adjacent to θ (not the hypotenuse) is called the "adjacent" side.
• The side opposite the right angle is the "hypotenuse".

Sine (sin)

• Sine of angle θ is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
• sin(θ) = Opposite / Hypotenuse

Cosine (cos)

• Cosine of angle θ is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
• cos(θ) = Adjacent / Hypotenuse

Tangent (tan)

• Tangent of angle θ is defined as the ratio of the length of the opposite side to the length of the adjacent side.
• tan(θ) = Opposite / Adjacent
• Tangent can also be expressed as sin(θ) / cos(θ).

Reciprocal Trigonometric Ratios

• Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocal trigonometric ratios.

Cosecant (csc)

• Cosecant of angle θ is the reciprocal of sin(θ).
• csc(θ) = Hypotenuse / Opposite = 1 / sin(θ)

Secant (sec)

• Secant of angle θ is the reciprocal of cos(θ).
• sec(θ) = Hypotenuse / Adjacent = 1 / cos(θ)

Cotangent (cot)

• Cotangent of angle θ is the reciprocal of tan(θ).
• cot(θ) = Adjacent / Opposite = 1 / tan(θ) = cos(θ) / sin(θ)

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables.
• They are helpful for simplifying expressions, solving equations, and proving other identities.

Pythagorean Identity

• The most fundamental trigonometric identity is derived from the Pythagorean theorem.
• sin²(θ) + cos²(θ) = 1
• This identity can be rearranged to isolate sine or cosine:
  • sin²(θ) = 1 - cos²(θ)
  • cos²(θ) = 1 - sin²(θ)

Other Pythagorean Identities

• Dividing the fundamental identity by cos²(θ) gives:
  • tan²(θ) + 1 = sec²(θ)
• Dividing the fundamental identity by sin²(θ) gives:
  • 1 + cot²(θ) = csc²(θ)

Angle Sum and Difference Identities

• These identities express trigonometric functions of sums and differences of angles in terms of trigonometric functions of the individual angles.
• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
• tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Double Angle Identities

• These identities express trigonometric functions of twice an angle in terms of trigonometric functions of the angle.
• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
• tan(2θ) = 2tan(θ) / (1 - tan²(θ))

Half Angle Identities

• These identities express trigonometric functions of half an angle in terms of trigonometric functions of the angle.
• sin(θ/2) = ±√((1 - cos(θ)) / 2)
• cos(θ/2) = ±√((1 + cos(θ)) / 2)
• tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ))) = sin(θ) / (1 + cos(θ)) = (1 - cos(θ)) / sin(θ)

Product-to-Sum Identities

• These identities convert products of trigonometric functions into sums/differences.
• sin(A)cos(B) = 1/2 [sin(A + B) + sin(A - B)]
• cos(A)sin(B) = 1/2 [sin(A + B) - sin(A - B)]
• cos(A)cos(B) = 1/2 [cos(A + B) + cos(A - B)]
• sin(A)sin(B) = 1/2 [cos(A - B) - cos(A + B)]

Sum-to-Product Identities

• These identities convert sums/differences of trigonometric functions into products.
• sin(A) + sin(B) = 2 sin((A + B)/2) cos((A - B)/2)
• sin(A) - sin(B) = 2 cos((A + B)/2) sin((A - B)/2)
• cos(A) + cos(B) = 2 cos((A + B)/2) cos((A - B)/2)
• cos(A) - cos(B) = -2 sin((A + B)/2) sin((A - B)/2)

Trigonometric Values for Common Angles

• Specific angles have trigonometric values that are frequently used.
• 0°, 30°, 45°, 60°, and 90° (or 0, π/6, π/4, π/3, and π/2 radians) are common.

Values at 0° (0 radians)

• sin(0°) = 0
• cos(0°) = 1
• tan(0°) = 0

Values at 30° (π/6 radians)

• sin(30°) = 1/2
• cos(30°) = √3/2
• tan(30°) = 1/√3 = √3/3

Values at 45° (π/4 radians)

• sin(45°) = 1/√2 = √2/2
• cos(45°) = 1/√2 = √2/2
• tan(45°) = 1

Values at 60° (π/3 radians)

• sin(60°) = √3/2
• cos(60°) = 1/2
• tan(60°) = √3

Values at 90° (π/2 radians)

• sin(90°) = 1
• cos(90°) = 0
• tan(90°) = undefined

Law of Sines

• The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles.
• For a triangle with sides a, b, c and angles A, B, C (opposite to the sides respectively):
  • a / sin(A) = b / sin(B) = c / sin(C)
• Can be used to find unknown angles or sides when given enough information (e.g., two angles and one side, or two sides and one angle opposite one of them).

Law of Cosines

• The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)
• Useful for finding unknown sides or angles when given three sides (SSS) or two sides and the included angle (SAS).

Applications of Trigonometry

• Used extensively in navigation, surveying, and astronomy.
• Essential for solving problems related to heights and distances.
• Used for modeling periodic phenomena like sound waves and light waves.